Iitaka's subadditivity conjecture C(n,m), with its logarithmic form and Campana's orbifold form

For a surjective morphism $f:X\to Y$ with connected fibres between smooth projective complex varieties, with general fibre $F$ , Iitaka's conjecture $C_{n,m}$ asks whether $\kappa(X)\ge\kappa(F)+\kappa(Y)$ . Iitaka also stated a logarithmic version for open varieties, $\bar\kappa(U)\ge\bar\kappa(F)+\bar\kappa(V)$ , and Campana conjectured an orbifold form in which rational boundaries and multiple fibres enter through an orbifold base: $\kappa(X,K_X+\Delta)\ge\kappa(F,K_F+\Delta_F)+\kappa(f\mid\Delta)$ for compact manifolds in Fujiki class $\mathcal C$ . Known cases included curve bases and fibres with good minimal models (Kawamata), general-type bases (Viehweg) and fibres (Kollar), total dimension at most six (Birkar), and abelian or maximal-Albanese bases. Does subadditivity hold for every such fibration?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Birational geometry, Kodaira dimension
Posed by: Shigeru Iitaka (J. Math. Soc. Japan 1971; Classification of Algebraic Varieties, Proc. Japan Acad. 1977); orbifold form by F. Campana (J. Inst. Math. Jussieu 2011, Conj. 6.1)
Year posed: 1977
Years open: 49y
Solved: 2026-09-26
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 62 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for a smooth compact connected manifold $X$ in Fujiki class $\mathcal C$ with rational SNC boundary $\Delta$ (coefficient one allowed) and a fibration $f:X\to Y$ onto a normal compact complex space, $\kappa(X,K_X+\Delta)\ge\kappa(F,K_F+\Delta_F)+\kappa(f\mid\Delta)$ for a very general fibre. Corollaries: logarithmic subadditivity for compatible reduced-SNC pairs, and ordinary $\kappa(X)\ge\kappa(F)+\kappa(Y)$ for projective fibre spaces over any algebraically closed field of characteristic zero. Not covered: positive characteristic, real or non-SNC boundaries, and the variation refinement, which is a separate entry.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The orbifold paper is the family's foundation: the variation and additivity companions use its logarithmic subadditivity and adjoint comparison as inputs, and the September 27 companion gives a second, purely projective proof of the ordinary case over any algebraically closed field of characteristic zero. The family is not formalized except for one branch of the additivity companion.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the introduction of 'Orbifold and logarithmic Iitaka subadditivity', read against Campana's Conjecture 6.1 as the paper quotes it and against Iitaka's ordinary and logarithmic inequalities, which it derives as corollaries (ordinary over any algebraically closed field of characteristic zero via geometric generic fibres); the September 27 companion's Theorem 1.1 states the ordinary case with an independent projective proof. The proofs (relative Iitaka reduction, root covers, Hodge-line positivity, general-type addition) were not refereed. No Lean formalization. The papers note earlier preprint claims of ordinary subadditivity by Tsuji and Maehara and say neither is an input. A reader should know that two later family-034 manuscripts (4 and 5 October) state logarithmic or orbifold Iitaka subadditivity as an explicit assumption; the 5 October one identifies that assumption with Theorem 1.1 of this paper, so the release does not contradict this claim.

Iitaka's subadditivity conjecture C(n,m), with its logarithmic form and Campana's orbifold form

What was actually shown

What the AI did

Verification

Sources

Discussion